Shift of argument subalgebras in Poisson algebras and their quantization.
Event details
| Date | 03.03.2015 |
| Hour | 15:15 › 17:00 |
| Speaker | Leonid Rybnikov, HSE Moscow |
| Location | |
| Category | Conferences - Seminars |
The symmetric algebra S(g) of a Lie algebra g carries a natural
Poisson bracket. Shift of argument subalgebras (introduced by Fomenko
and Mishchenko in 1978) form a family of maximal Poisson-commutative
subalgebras in S(g) for semisimple g. This family is parametrized by
regular elements of the dual space g*. I will discuss the quantization
problem for shift of argument subalgebras, namely, how to lift these
subalgebras to commutative subalgebras in the universal enveloping
algebra U(g), and how to describe the spectra of the "quantum shift of
argument subalgebras" of U(g) on (finite-dimensional) g-modules. These
questions are related to the classical representation theory, in
particular, it was observed by Vinberg, that the Gelfand-Tsetlin
subalgebra in U(gl_n) is a certain limit of quantum shift of argument
subalgebras. Hence the spectra of quantum shift of argument
subalgebras on a finite-dimensional gl_n-module can be regarded as a
deformation of the corresponding Gelfand-Tsetlin polytope.
The construction of the quantum shift of argument subalgebras is a
version of the Feigin-Frenkel-Reshetikhin construction of higher
hamiltonians for the Gaudin model. The
quantum shift of argument subalgebras come from the center of the
universal enveloping algebra of the corresponding affine Lie algebra
g^ at the critical level by an appropriate quantum Hamiltonian
reduction. The center at the critical level is naturally identified
with the algebra of polynomial functions on the space of opers on the
formal punctured disk with respect to the Langlands dual group G^L
(roughly, the space of gauge equivalence classes of connections in a
principal G^L-bundle with some transversality condition). This allows
us to treat the spectra of the quantum shift of argument subalgebras
on g-modules as some subsets in the space of opers. I will give a
precise description of these subsets for irreducible
finite-dimensional g-modules.
Poisson bracket. Shift of argument subalgebras (introduced by Fomenko
and Mishchenko in 1978) form a family of maximal Poisson-commutative
subalgebras in S(g) for semisimple g. This family is parametrized by
regular elements of the dual space g*. I will discuss the quantization
problem for shift of argument subalgebras, namely, how to lift these
subalgebras to commutative subalgebras in the universal enveloping
algebra U(g), and how to describe the spectra of the "quantum shift of
argument subalgebras" of U(g) on (finite-dimensional) g-modules. These
questions are related to the classical representation theory, in
particular, it was observed by Vinberg, that the Gelfand-Tsetlin
subalgebra in U(gl_n) is a certain limit of quantum shift of argument
subalgebras. Hence the spectra of quantum shift of argument
subalgebras on a finite-dimensional gl_n-module can be regarded as a
deformation of the corresponding Gelfand-Tsetlin polytope.
The construction of the quantum shift of argument subalgebras is a
version of the Feigin-Frenkel-Reshetikhin construction of higher
hamiltonians for the Gaudin model. The
quantum shift of argument subalgebras come from the center of the
universal enveloping algebra of the corresponding affine Lie algebra
g^ at the critical level by an appropriate quantum Hamiltonian
reduction. The center at the critical level is naturally identified
with the algebra of polynomial functions on the space of opers on the
formal punctured disk with respect to the Langlands dual group G^L
(roughly, the space of gauge equivalence classes of connections in a
principal G^L-bundle with some transversality condition). This allows
us to treat the spectra of the quantum shift of argument subalgebras
on g-modules as some subsets in the space of opers. I will give a
precise description of these subsets for irreducible
finite-dimensional g-modules.
Practical information
- General public
- Free